信息复杂性与腐败及其在随机性和间隙汉明中的应用
Information Complexity versus Corruption and Applications to Orthogonality and Gap-Hamming
摘要：
三十年来对通信复杂性的研究导致了许多技术的发明，以降低随机通信复杂性。这些技术中的大多数涉及定义通信问题的真值表矩阵的大型子矩阵（矩形）的属性。唯一不太适合的技术是信息复杂性，它在过去十年中一直在研究。在这里，我们将信息复杂性与最强大的“矩形”技术之一联系起来：最近引入的平滑腐败（或“平滑矩形”）界限。我们表明，前者包含后者矩形输入分布。我们推测，这种包容持有更普遍的，在任意分布，这将解决长期以来的直接和问题的随机通信。作为应用，我们得到了一个最佳的$Ω(n)$下界的信息复杂度-下的{ \em均匀分布}-所谓的正交性问题（ORT），这又是密切相关的研究间隙汉明距离（GHD）。这个界限的证明是沿着最近GHD的通信下限的路线，但我们遇到了令人惊讶的额外的技术细节。
英文摘要：
Three decades of research in communication complexity have led to the invention of a number of techniques to lower bound randomized communication complexity. The majority of these techniques involve properties of large submatrices (rectangles) of the truth-table matrix defining a communication problem. The only technique that does not quite fit is information complexity, which has been investigated over the last decade. Here, we connect information complexity to one of the most powerful "rectangular" techniques: the recently-introduced smooth corruption (or "smooth rectangle") bound. We show that the former subsumes the latter under rectangular input distributions. We conjecture that this subsumption holds more generally, under arbitrary distributions, which would resolve the long-standing direct sum question for randomized communication. As an application, we obtain an optimal $Ω(n)$ lower bound on the information complexity---under the {\em uniform distribution}---of the so-called orthogonality problem (ORT), which is in turn closely related to the much-studied Gap-Hamming-Distance (GHD). The proof of this bound is along the lines of recent communication lower bounds for GHD, but we encounter a surprising amount of additional technical detail.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
